introduce the Median Radial Basis Functions (MRBF) algorithm based on robust .... P roviding their capabilities of approximation [1]- [3] , RBF networks can be ...
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1 X(1)
Y1
2
X(2)
Y2
YM
X(N)
L INPUT
HIDDEN LAYER
8.
G»
»
OUTPUT
0.08 p 2
0.07 0.06
p 3 0.05
p 1
0.04 0.03 0.02 0.01 0 0
2
4
T1 6
8
8.
10 T2
12
14
16
0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06
E ? `A ?a A
0.04
0.02 0 0
2
4
6
8
10
8.XN!
N
12
14
16
-0.02
Median
Expected bias of the location parameter
-0.04 -0.06 -0.08 -0.1 -0.12 -0.14 -0.16 -0.18
Mean
-0.2 -0.22
0
20
40
60
80
100
120
140
160
180
200
140
160
180
200
No. samples
Y. /mrk?@ A
0.1
Expected bias of the scale parameter
0.05 0 -0.05 MAD
-0.1 -0.15 -0.2 -0.25 -0.3
Sample standard deviation 0
20
40
60
80
100 No. samples
8.mr?S#pA 8.mr
r
120
Bias in center estimation
-0.5 -- mean -1 - median -1.5 1
2 3 Scale parameter (a)
4
0
Bias in scale estimation
Bias in center estimation Bias in scale estimation
0
-0.5
-1
-- classical - MAD
-1.5 1
2 3 Scale parameter (c)
4
8.Xx!
x
0.8 0.6 0.4
-- mean
0.2
- median
0 1
2 3 Scale parameter (b)
4
0
-1 -- classical
-2
- MAD -3
1
2 3 Scale parameter (d)
4
20
15
6
10
FEATURE II
. . ... . . .. .. . . . . ... .. . ... ... . . . . . . . . . .. .. . . .. . .. .. . . . . . . . .. . . ... . . ... . . . . . . ...... .... ..... .... . . . . . . .. . . .. ... . .. . . . . . . . . . . . . . . . . . . . .. . .. . . .... . . . . .... . . ... . ..... ......... ........... ......... .... . .. .. ........... ... .. .... .. ..... .. .... . ...... . . . . . . .. . . . . . . . .. . . .. . . . .. .... ...... ... ... . . ... .... ...... .. .. ... ...... .... . .. ... .. .. .. .. . . . . .. . .... .. ..... . .... ..... .. .......... ..... ........ .. .... .... ..... . ........ . . . . . .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ... .. .. . . . . ... . ........ .. . . . . .. .. . . . ... . . . . . . ... . ..................... ........ .. . ........ ........... .... ... ..... ... .. .. .... . . . ..... ... . . ... . ... . . ... .. .. .... . . .. . . . . .. .... .... ... .. .. . . . . . . . ... ... . .. . . .. ... ... . ............ . .. ......... ... .. .... ........ . . . .. . . . . . ... . . ... ... ..... . . .. .. .. ... . . . . . . . ... . . ... .. ........ .... ..... ........ ........ . ....... ... ... . .. .......... .. .......... . . .. . . . . . . .. . . . . . . . ........ ....... ........... ... ... ......... . . .. .... .... .. . .. .... . . ... . ..... . . . .. . .. .... . ........... ... ... ..... .......... ..... ............ ..... ... ......................... .. ... . . . . . .. . ...... .... ... .... .. ......... .. ...... .. . ..... . .. ... ............................ ... .. . . . . . . . . . . .. ..... ... . . . . . . . . .. .. .. . . . . ... . .. . . . ...... .. . .. ... .. . . .. .. . . . . . . .... . . . . . . . . .. .... . . . . . . .
5
.. .
.
0
.
. . . . . .. .. .
6E
6E
-5
.
6
.
.
.
-10 -10
-5
0
5 FEATURE I
8.X!
10
15
20
20 .
15
6
FEATURE II
10
5
0 .
-5
. .. . . . .. . . . . .. . .... ..... .. . . . . . ..... . ... . . ..... . . . . . .... ..... ... . .. . . . . .. .. . .. . . . . . . . . . . . . . . ............. ........ .. . . . . . . . . . . . . .. ...... .... .. . . . .. .... . . .. . . . .... . . . .. . ..... ... ........................ ........ . . . .. .. . ... .... ... .. ... ...... .. . ... ... . .. .... . .......... .... ...... .... . ...... ... .. . . .... ... .. .. .. . . ... . . . .. . . .. .... ............... ..... . . . .. . .. . .. ........ ....... .... ... .. ...... ... . . . ... . . . . .. .... ............................................. .. ...... .. .. .... ............... .. ................................ ......... ... . . . . . . . . . . ......... .................................................. ........... ... .......... ... .............................................. ... ... ...... . ... ... . . .... ........................................ ....... .... . . .......... ....... ............... . ................................................. ................ ... . .. .... . . . . . ................ ................. .. .. . . .... .. .. ........................................... .. .................... . ... . ... .. .. . .. . . . . . . . ........ ............................... ... .. . ... . .. ....... ... .............. ....... ..... ... .. ... ...... .. . . . .. . ....... ................................ . . . . . .... ............... .... ........ .. ............ ......... .. . ... . . . . . . . . . . ..... . .. . . . . . . . . . . . .. . . .. . . . . . . . . . .. . ... .......................... ..... ... . ....... . . .. ... ....... .. . ... ......... .. . .... .. . ... . . ..... .... ....... . . . . . . ... . . ......... ..... .. .. .. .... .... . ... .. . . .. . .. . . . . . . . . ... . .. . .. . ..... ....... ... .... . ........ .. . .... .. .. .............. ... . ...... . . .. . .. . . . . . . . .. . ............................. ........................................................................................... ..................... ....... ............ ... .. .. . ..... . . .. . . . . . .. .... .. ... .. ......... ................ .... .. ................... ....................................... ... ...... ............................................ .... . ..... . .... . . . ... ....... ... . ....... . . . . ... .... . .. .. .................. .. ... .... ..... ......... ..... . .. . ... . .... . . . . . . . . . . . . . ..... . . ... ......... .............. . ... . ............ . .. .... .. .......... ... ...... .... .. .. ......... .... . .... . ..... .... . . . .. . . ... .. . . . .. ... . . . .. . . . . ... ... . . . . . . . . . . . . . . . . .. .. . . .. . ...
6E
. .
. .
. .
-10 -10
-5
.
0
5 FEATURE I
8.UE
10
15
20
25
Mean Square Error
20
15
RBF - Euclidean
10 MRBF - Euclidean RBF - Mahalanobis 5 MRBF - Mahalanobis 0
0
500
1000
1500
2000 Data samples
8.X!
2500
3000
3500
4000
Y. /X|?@ A
8.X?S#pA 8.X!
8$. %» ?e A
8. 0» ?S#pA
N
8. 0» ?@A Y. / 0»
N
»
n
0 Y
X 0
n
8$.
»V»
?e A
8.
»»
?S#pA
0 Y
X 0
N
n
0 Y
X 0
8. Y. /
N N
»» »»
?@A